The method of proving this depends mostly on the structure of the target and only minimally on that of the source. It is not hard to generalise it to manifolds with boundary (to get, for example, path spaces), or even manifolds with corners. This raises the obvious question as to how general this result can be made. The purpose of this page is to determine the answer. Our conjecture is the following:

Construction of smooth manifold structure on mapping space

The following needs attention. For a more recent version see (Stacey).

Background and Remarks

The question discussed here can be viewed as the counterpoint to the oft-heard maxim (attributed to Grothendieck):

It is better to work in a nice category with nasty objects than in a nasty category with nice objects.

Smooth manifolds are an example of “nice objects in a nasty category”; for example, one can rarely take subobjects or quotients. The standard procedure at this point is to embed the nasty category in some larger, nicer category and work there. In the case of smooth manifolds, this has led to all of the categories that are listed at generalized smooth space.

One can now go on to study this enlarged category, and investigate how much of what is known about the original category extends to the larger one. In this line, the original category is viewed mainly as a source of ideas. An alternative approach, and that taken here, is to view the original category as being a subcategory of “special objects” inside the larger one.

One can make an analogy with the real and complex numbers. Many aspects of the study of real numbers become much easier and clearer when extended to the complex numbers. At this point, one has a choice: one can simply study the complex numbers or one can use the complex numbers as a tool to study the real ones.

Thus, to adapt a saying of Hadamard, we could introduce our own maxim:

The shortest distance between two truths about nice objects often lies in a nice category.

Having mentioned the plethora of extensions of the category of smooth manifolds, we should comment on our choice of Frölicher spaces. The inclusion of the category of smooth manifolds into each of the extensions factors through the category of Frölicher spaces. Therefore, if we work in, say, the category of diffeological spaces then we can split the question “Is the diffeological space XX a smooth manifold?” into “Is XX a Frölicher space?” and “Is the resulting Frölicher space a manifold?”. Moreover, as we are interested in C∞(N,M)C^\infty(N,M) with MM a smooth manifold (and thus a Frölicher space), then if we are working with one of the “maps in” approaches, we can replace the NN in C∞(N,M)C^\infty(N,M) by its “Frölicherification” without changing the set. Thus the key piece of the puzzle is to study C∞(N,M)C^\infty(N,M) for NN a Frölicher space and the rest will follow by applying “general nonsense”.

Another remark worth saying is that the conjecture stated is not the most general statement that could be considered. It is simple to extend this conjecture to a relative version whereby MM is equipped with a family of submanifolds and NN with a family of subsets and the maps are constrained to take the subsets to the corresponding submanifolds.

Finally, let us note that the main results about the linear model spaces are recorded on the page linear mapping spaces.

Definition

We write C∞(N,M;Qi,Pi)C^\infty(N,M;Q_i,P_i) for the set of smooth functionsN→MN \to M which map each QiQ_i into the corresponding PiP_i.

As a smooth manifold, MM naturally has the structure of a Frölicher space so this mapping space is well-defined.

We assume that the pair(M,{Pi})(M,\{P_i\}) admits a local addition. By that, we mean that MM admits a local addition, say η\eta, with the property that it restricts to a local addition on each PiP_i. We shall also assume, for simplicity, that the domain of η\eta is TMT M.

Let g:N→Mg \colon N \to M be a smooth map with g(Qi)⊆Pig(Q_i) \subseteq P_i. Let EgE_g be the space of sections of g*TMg^* T M with the property that the sections over each QiQ_i are constrained to lie in the corresponding g*TPig^* T P_i. In more detail, we define g*TMg^* T M in the usual manner:

and then take the space of smooth maps f:N→g*TMf \colon N \to g^* T M with the property that the composition N→g*TM→NN \to g^* T M \to N is the identity. Within that space, we further restrict to those ff such that the image of the map Qi→g*TM→TMQ_i \to g^* T M \to T M lies in TPiT P_i.

Although NN could be quite complicated, because TM→MT M \to M is a vector bundle, EgE_g is a vector space. Furthermore, by trivialising g*TMg^* T M using a finite number of trivialisations (possible as NN is sequentially compact), we can embed EpE_p as a closed subspace of C∞(N,Vn)C^\infty(N,V^n) for some nn. This embedding shows that EpE_p is a convenient vector space, in the sense of Kriegl and Michor.

Andrew Stacey This, I think, is the crucial part: that EpE_p is a convenient vector space. I need to expand on this and check that all is as I think it is.

We define a map for Φ:Eg→C∞(N,M;{Qi},{Pi})\Phi \colon E_g \to C^\infty(N,M;\{Q_i\},\{P_i\}) as follows. Let f∈Epf \in E_p. Then ff is a section of g*TMg^* T M and so is a map N→g*TMN \to g^* T M. By the definition of g*TMg^* T M, we can think of ff as a map N→N×TMN \to N \times T M which projects to the identity on the first factor. By applying the projection to the second factor, we obtain a map f^:N→TM\hat{f} \colon N \to T M. Composing with η\eta produces a map η∘f^:N→M\eta \circ \hat{f} \colon N \to M. As f∈Egf \in E_g, the restriction of f^\hat{f} to QiQ_i lands in TPiT P_i, whence η∘f^\eta \circ \hat{f} takes QiQ_i into PiP_i. The map f↦η∘f^f \mapsto \eta \circ \hat{f} is what we call Φ\Phi.

Let us identify its image. Let V⊆M×MV \subseteq M \times M be the image of the local addition. Define Ug⊆C∞(N,M;{Qi},{Pi})U_g \subseteq C^\infty(N,M;\{Q_i\},\{P_i\}) to be the set of those functions hh such that (g,h):N→M×M(g,h) \colon N \to M \times M takes values in VV. We claim that the image of Φ\Phi is UgU_g and that Φ\Phi is a bijection Eg→UgE_g \to U_g.

Let us start with the image. Let h∈Ugh \in U_g. Then (g,h):N→M×M(g,h) \colon N \to M \times M takes values in VV, so we can compose with (π×η)−1(\pi \times \eta)^{-1} to get a map hˇ:N→TM\check{h} \colon N \to T M. Together with the identity on NN, we get a map N→N×TMN \to N \times T M. By construction, πhˇ=g\pi \check{h} = g and so this map ends up in g*TMg^* T M (which has the subspace structure). Again by construction, the projection of this map to NN is the identity and so it is a section of g*TMg^* T M. That it takes QQ to TPT P follows from the fact that η\eta restricts to a local addition on PP, whence as h(Q)⊆Ph(Q) \subseteq P, hˇ(Q)⊆TP\check{h}(Q) \subseteq T P. Hence Φ\Phi is onto. Moreover, this construction yields the inverse of Φ\Phi and so it is a bijection.

Thus we have charts for C∞(N,M;Q,P)C^\infty(N,M;Q,P).

Transition functions

The next step is the transition functions. To prove this in full generality, we assume not just two different functions at which to base our charts, but also two different local additions to define them. This will show that our resulting manifold structure is independent of this choice. We could go further than we do, and allow our local additions to be in the most general form given at local addition, but this would crowd the notation with little benefit.

where we have used the fact that (x,v)∈g1*TM(x,v) \in g_1^*T M so π(v)=g1(x)\pi(v) = g_1(x). Thus ϕ21\phi_{2 1} and ϕ12\phi_{1 2} are inverses, whence they are diffeomorphisms.

Lemma

The transition function is f↦ϕ21∘ff \mapsto \phi_{2 1} \circ f.

Proof

Let us start with the domain and codomain of the transition function. The domain is {f∈E1:Φ1(f)∈U2}\{f \in E_1 : \Phi_1(f) \in U_2\}. The set U2U_2 consists of those functions h:N→Mh \colon N \to M such that (g2,h)(g_2, h) takes values in V2V_2. Thus Φ1(f)∈U2\Phi_1(f) \in U_2 if and only if (g2,Φ1(f))∈V2(g_2, \Phi_1(f)) \in V_2. Since Φ1(f)=η1∘f^\Phi_1(f) = \eta_1 \circ \hat{f}, we see that for x∈Nx \in N, v≔f^(x)∈TMv \coloneqq \hat{f}(x) \in T M must be such that (g2(x),η1(v))∈V2(g_2(x), \eta_1(v)) \in V_2. This is precisely the condition that (x,v)(x,v) be in W12W_{1 2}. Thus the domain of the transition function is the set of sections f∈E1f \in E_1 such that f(x)∈W12f(x) \in W_{1 2} for each x∈Nx \in N.

The transition function, Ψ21\Psi_{2 1}, is given by Ψ21=Φ2−1Φ1\Psi_{2 1} = \Phi_2^{-1} \Phi_1. It is therefore completely characterised by the fact that Φ2Ψ21=Φ1\Phi_2 \Psi_{2 1} = \Phi_1.

Let us consider Φ2\Phi_2 applied to ψ21∘f\psi_{2 1} \circ f for f∈E1f \in E_1 such that ff takes values in W12W_{1 2}. Expanding out the definition, we have: